Nearest just interval

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An irrational interval or ratio of frequencies given by a real number r has an infinite list of nearest just intervals; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call best rational approximations. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but |4/3 - √2| = .08088 whereas |3/2 - √2| = 0.08479, which is larger.

Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest.

The semiconvergents of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely best relative approximation. Here it is required that |qr - p| is less than |nr - m| for any n < q.

Examples

Approximations for Ratios (of Pure Intervals)

The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... cents):

Step\EDO log(Tenney Height) size in cents "error" in cents
... ... ... ...
1 \ 1 0.0 1200.0 498.04
1 \ 2 1.0 600.00 -101.96
2 \ 3 2.585 800.00 98.045
3 \ 5 3.907 720.00 18.045
4 \ 7 4.807 685.7143 -16.2407
7 \ 12 6.392 700.00 -1.955
17 \ 29 8.945 703.4483 1.4933
24 \ 41 9.943 702.43902 0.48402
31 \ 53 10.682 701.88679 -0.06821
  • for approximations of the harmonic seventh see 7_4

Approximation for Logarihmic Measures

The 600-cent interval sqrt(2) (6 steps of 12edo, "Tritone") approximates following ratios:

freq. ratio log2(Tenney Height) size in cents "error" in cents
... ... ... ...
1 / 1 0.0 0.0 600.0
3 / 2 2.585 701.96 101.96
4 / 3 3.585 498.04 -101.96
7 / 5 5.129 582.51 -17.49
17 / 12 7.672 603.000 3.000
24 / 17 597.000 -3.000
99 / 70 600.0883 0.0883
140 / 99 599.9117 -0.0883
... ... ... ...

The 300-cent interval 2^(1/4) (3 steps of 12edo, "minor third") approximates following ratios:

freq. ratio log(Tenney Height) size in cents "error" in cents
... ... ... ...
1 / 1 0.0 0.0 300.0
6 / 5 4.907 315.64 15.64
13 / 11 7.160 289.21 -10.79
19 / 16 8.248 297.51 -2.49
25 / 21 9.036 301.84 1.84
... ... ... ...